I’m trying to prepare for further study in Analysis and was wondering what advice you all would give. I have read (most of) Abbott’s Understanding Analysis and have started Rudin’s Principles of Mathematical Analysis. I plan to finish Abbott within the next couple of weeks and also Rudin by the middle of June or early July. Would this be adequate preparation for Royden’s Real Analysis or (preferably) Folland’s Real Analysis? Note: I have taken set theory, a 300-level linear algebra course, and will be taking point set topology this upcoming summer.

I say that Folland is preferable because I would, so long as everything goes well, like to take my university’s graduate real analysis sequence, and Folland is the assigned text. I would like to advance to this level because I’m not sure that my school will offer the second semester of their senior-level real analysis sequence – they don’t always.

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I haven't read Folland, but I have read both Royden's book and Rudin's Real & Complex Analysis. Royden's book, at least the edition at the time, had one or two pretty serious gaps in proofs. It did cover some territory Rudin didn't cover (understandably, since only the first half of Rudin is about real analysis), but Rudin was impeccably correct. I would not suggest Royden.

In terms of prep work, you'll have done plenty if you read the prep books you are planning to read and fully digest them.